$\newcommand{\dis}{\displaystyle} \newcommand{\m}{\hspace{1em}} \newcommand{\mm}{\hspace{2em}} \newcommand{\x}{\vspace*{1ex}} \newcommand{\xx}{\vspace*{2ex}} \let\limm\lim \renewcommand{\lim}{\dis\limm} \let\fracc\frac \renewcommand{\frac}{\dis\fracc} \let\summ\sum \renewcommand{\sum}{\dis\summ} \let\intt\int \renewcommand{\int}{\dis\intt} \newcommand{\sech}{\text{sech}} \newcommand{\csch}{\text{csch}} \newcommand{\Ln}{\text{Ln}} \newcommand{\p}{\partial} \newcommand{\intd}[1]{\int\hspace{-0.7em}\int\limits_{\hspace{-0.7em}{#1}}} $
Lecture 37, 11/28/2022.
This page is for Section 1 only
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ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A
Chapter 7: Fourier Series and Transform
An application to sound:
If an amplitude of a frequency is much larger than that of the others, that is the frequency you can hear.
Don't forget
for $\sin (\omega t)$ or $\cos (\omega t)$,
frequency$=\frac\omega{2\pi}$
Parseval's theorem:
$ \frac1{2l}\int_{-l}^l |f(x)|^2 dx = \Big(\frac12 a_0\Big)^2 + \frac12\sum_{n=1}^\infty a_n^2 + \frac12\sum_{n=1}^\infty b_n^2$